The certificate and the license are two different things.
The license is your permit from the state to marry. It does not mean that you are married once you apply for it. It means that you have legal permission to marry.
You need to apply for the license in advance of your wedding - at the VERY least, a week or two in advance. Applying at least a month or two out would be wise, especially if either of you has been married prior to this. They need time to process the paperwork ... many states take at least three business days (weekdays, not weekends or holidays) from the application date to issue you a marriage license. Vegas is one of the rare exceptions where you can get it the same day you plan to marry.
You need to pick up the license before the wedding day, so that your officiant and witnesses can sign it once the ceremony is over.
You need to ask the clerk when you should pick it up. Validity varies by state. In my state, you need to use it within six months (I forget if that countdown starts on the date you apply, the date they process it, or the date you pick it up). We applied about two months out and picked it up a couple weeks prior to the wedding. We needed our photo IDs, a witness age 18+, and a cash fee in order to apply for the license.
The certificate is what you receive after you're married - it is legal proof that you are legally married. You only get that after your wedding. The officiant and your witness sign your license once the ceremony is over, the officiant mails it to the proper authorities, they process it, and then they will send you a certified copy of your marriage certificate. This is what you use as legal proof that you're married, and if you want to change your last name then this is the document you need.